Let $(M,g)$ be a Riemannian manifold and $V$ a unit Killing vector field on it. Under what condition on curvature tensor the following equation hold: $$\nabla_VQ=0,$$ where $Q$ is the Ricci operator defined as $g(QX,Y)=\rho(X,Y)$.

**Update1:** Einstein metrics that admit a unit Killing vector field satisfies the above equation.

**Update2:** Can anybody give an example other than Einstein manifolds that satisfy the above equation?